GREENVILLE, S.C. — He’s a rock star to many Republicans, particularly in South Carolina, but Rep. Trey Gowdy appears to be a stranger to Ohio Gov. John Kasich, who called him “Troy Gowdy” on Tuesday as he campaigned in Mr. Gowdy’s congressional district.

Having just flown into South Carolina after campaigning in Michigan, Mr. Kasich is playing catch-up with the rest of the Republican presidential field, and his rustiness showed as he fielded questions at a town hall.

“I don’t know where I am, I don’t know where I’m going,” the governor said, laughing at the hectic pace he is keeping as he tries to turn last week’s surprise second-place showing in New Hampshire’s primary into a broader movement.

Still, he drew repeated rounds of applause from an audience that seemed appreciative of his unwillingness to pander, telling them that there are some problems local communities need to solve, rather than turning to the federal government.

At one point, he was asked whether he supported a bill to improve interactions between police and the mentally ill, which a questioner said was sponsored by Mr. Gowdy and is awaiting action by the full U.S. House of Representatives. Mr. Kasich said he supports help for the mentally ill and that it was a big reason he expanded Medicaid under Obamacare in his state, but he wouldn’t commit to supporting the bill in Congress without having read it.

Minutes later, he fielded a question from a woman who said her father is having trouble accessing the benefits he earned from his time in the military. Mr. Kasich pointed back to the man who inquired about mental health and asked him, “You know Troy Gowdy?” — apparently hoping the man could connect the woman with the local congressman. The audience quickly corrected Mr. Kasich’s error, shouting back, “Trey.” Mr. Kasich then dispatched his own staffer to talk with the woman, who broke into sobs as she explained her difficulty in getting her father’s benefits.

Mr. Gowdy has gained national attention by leading the congressional investigation into former Secretary of State Hillary Clinton’s emails from her time in office. Mr. Gowdy is supporting Sen. Marco Rubio of Florida, one of the other candidates in the presidential race.

Mr. Kasich is banking his campaign on being less confrontational than the rest of the Republican field. Audiences for Sen. Ted Cruz of Texas and businessman Donald Trump often push for ideological purity, but Mr. Kasich’s audiences appear to respond to his stance as a compromiser.

“I’m going to do my best to stay relentlessly positive,” he said in kicking off his wide-ranging remarks.

He said the key to forging compromise in Washington is to know the parents of members of Congress. Mr. Kasich said he would get his way by making sure to call lawmakers’ parents on their birthdays: “Happy birthday, Mom, this is the president. And, by the way, can you tell your kid to help me?”